I think we'll all agree that, even if no Scorpions went with the Dragoons, it would make sense for a little bit of Scorpionish Tendencies to rub off.
"That is enough training today. You have done well."
"Excuse me, Star Commander Hank Scorpion*, but may I ask what is that book you're reading?"
"Why of course, Cranston Wolf. It is a copy of Game of Star Lords by Martin RR George**, the Star League's best selling book of the year 2652. It is a novel about feuding branches of the Terran Hegemony's ruling family, and their interactions with the leaders of the other members of the Successor States. A bit long, but a good read nonetheless."
"...where did you get that?"
"Oh, dug it out of a Brian Cache last Thursday. Had to fight a Star of rampaging Hellion Elementals while armed only with a spoon, but it was worth it."
"That is the coolest thing I've ever heard of in my life, and I want to model my life exactly that way."
"Here, you may need this bottle..."
While it probably wouldn't be anything remotely this silly, you see what I'm getting at. It would be natural for these groups working together to have some interests rub off on each other.
*Shut up. It's my example, I'll stick Simpsons references in if I want to, dammit.
**I really have no shame.